Vector Spaces ≡ Linear Spaces
A vector space over a field F is a set V together with two binary operations that satisfy eight axioms listed on this Wiki page. Basically, commutativity, associativity, identity and inverse of vector addition plus distributivity, identity etc of scalar multiplication.
A example in geometry of a vector space is arrows in a plane - adding vectors to form a parallelogram.
Another example are linear equations, e.g.:
a + 3b + c = 0
4a + 2b + 2c = 0
The coefficients can be turns into a matrix and (a, b, c) into a vector such that:
A x = 0
where A is the matrix, x the vector and 0 = (0, 0), the zero-vector.
Definition of a Topology
Let X be a set and T be a family of subsets of X. T is called a topology on X if:
∅ ∈ T and X ∈ T
Where ∀ α, Uα ⊂ T and α is potentially infinite:
∪αUα ∈ T (≡ T is stable by arbitrary union)
Where ∀ α, Uα ⊂ T and α is finite:
∪α∩α ∈ T (≡ T is stable by finite intersection)
Example 1
If X = {1,2,3,4,5}, then
T1 = {∅,{1,2},{3,4},{1,2,3,4},X}
and
T2 = {∅,{1,2},{2,3},{2},{1,2,3},X}
are both topologies on X.
Example 2
The trivial topology equals {∅,X} and the discreet topology equals { all subsets of X } ≡ 2X
These fulfil all the definitions of a topology (see above).
Definition of a Topological Space
… is the pair (X, T). So, in the above example, (X, T1) and (X, T2) are both topological spaces.
Definition of Open and Closed Sets
All elements of T are called open sets. The complement of all elements of T are called closed sets.
So, for any U ⊂ T which is open, its complement (X \ U) is closed.
Definition of the Usual Topology on ℜ
… is T where
O ∈ T iff ∀ x ∈ O ∃ ε > 0 : ] x − ε, x + ε [ ⊂ O
Definition of Normed Vector Space
∀ x ∈ E, N(x) = 0 ⇔ x = 0
∀ (x, λ) ∈ E x R, N(λ x ) = | λ | N(x)
∀ (x, y) ∈ E x E, N(x) + N(y) ≤ N(x + y)
"If you have all of this, E is a normed vector space."
If the first rule does not apply but the second and third do, then it's a seminorm space.
The Relationship between Spaces
The relation between spaces is explicitly encoded in the Breeze math library written in Scala. Here, we can see the InnerProductSpace trait extend the NormedVectorSpace trait that in turn extends the VectorSpace trait.
[1] An Introduction to Functional Analysis - Coursera.
